Download Second Level Numeracy

Kirkhill Primary School
Curriculum for Excellence
A Guide for Parents and Carers to Support Learning at Home
NUMERACY
&
MATHEMATICS
SECOND LEVEL
This booklet outlines the skills pupils will develop in Numeracy and Mathematics
within the Second Level.
This document makes clear the correct use of language and
agreed methodology for delivering Curriculum for
Excellence Numeracy and Mathematics experiences and
outcomes within the Mearns Castle Cluster. The aim is to
ensure continuity and progression for pupils which will
impact on attainment.
We hope you will find this booklet useful in helping you to
support your child at home.
Number and number processes
I have extended the range of whole numbers I can work with and having
explored how decimal fractions are constructed, can explain the link
between a digit, its place and its value.
Example
205
236
05
2145 - decimal fraction
- common fraction
2
Correct Use of Language
Say:
two point zero five, not two
point nothing five.
two point three six not two
point thirty-six.
zero point five not point five.
Talk about decimal fractions and
common fractions.
Methodology
Ensure the decimal point is placed
at
middle height.
Number and number processes
Having determined which calculations are needed, I can solve problems
involving whole numbers using a range of methods, sharing my
approaches and solutions with others.
I have explored the contexts in which problems involving decimal
fractions occur and can solve related problems using a variety of
methods.
Having explored the need for rules for the order of operations in number
calculations, I can apply them correctly when solving simple problems.
Example:
5 6
+ 31 9
9 5
2 6
×2 4
10 4
4 7
 54 6
2 8 2
3
2 31 5 0
2 6 3 2
Correct Use of Language
For multiplying by 10, promote the
digits up a column and add a zero
for place holder.
For dividing by 10, demote the
digits down a column and add a
zero in the units’ column for place
holder if necessary.
Methodology
When “carrying”, lay out the
algorithm as in the example.
Put the addition or subtraction sign
to the left of the calculation.
When multiplying by one digit, lay
out the algorithm as in the
example.
The “carry” digit always sits above
the line.
Decimal point stays fixed and the
numbers move when multiplying
and dividing.
Do not say, “add on a zero”, when
multiplying by 10. This can result
in 36 10=360.
Number and number processes
I can show my understanding of how the number line extends to include
numbers less than zero and have investigated how these numbers occur
and are used.
Term/Definition
Negative numbers
Example
4
0oC
Correct Use of Language
Say negative four not, minus four.
Pupils should be aware of this as a common mistake, even in the media
e.g. the weather.
Use minus as an operation for subtract.
Twenty degrees Celsius, not centigrade
Explain that it should be negative four, not minus four.
Fractions, decimal fractions and percentages
I have investigated the everyday contexts in which simple fractions,
percentages or decimal fractions are used and can carry out the
necessary calculations to solve related problems.
I can show the equivalent forms of simple fractions, decimal fractions
and percentages and can choose my preferred form when solving a
problem, explaining my choice of method.
I have investigated how a set of equivalent fractions can be created,
understanding the meaning of simplest form, and can apply my
knowledge to compare and order the most commonly used fractions.
Term/Definition
Methodology
Numerator: number above the line
Emphasise the connection between
in a fraction.
Showing the number of parts of the finding the fraction of a number
and its link to division (and
whole.
multiplication).
that the equivalence of 24
Denominator: number below the Ensure
1
and
is
highlighted. Use concrete1
line in a fraction.
2
examples
illustrate this. Show 4
The number of parts the whole is is smaller to
than 12 . Pupils need to
divided into.
understand equivalence before
introducing
other fractions such as
1
or 15 .
3
Example
Accept all common language in
245 decimal fraction
use:
1
Five pounds eighty ,
common fraction
2
Five pounds eighty pence,
Five eighty.
Start 9with 4 101 is written 41
7 10 is written 79 etc.
Ensure the decimal point is placed
at middle height.
37
Then 3 100
is written 337 etc.
To find 34 of a number, find one
Finally 6 34 is the same as
quarter first and then multiply by 3.
75
6 100
which is 675
Correct Use of Language
1
4
- Emphasise that it is one
divided by four.
£580 – Say five pounds eighty to
match the written form. DON’T
WRITE OR SAY £580p.
Say:
two point zero five, not two point
nothing five. (205)
two point three six, not two point
thirty-six.(2.36)
zero point five not, point five. (0.5)
Pupils should be aware of the
phrases state in lowest terms or
reduce.
Talk about decimal fractions and
common fractions to help pupils
make the connection between the
two.
Simplifying fractions – Say,
“What is the highest number that
you can divide the numerator and
denominator by?” Check by
asking, “Can you simplify again?”
Finding equivalent fractions,
particularly tenths and hundredths.
Teach fractions first then introduce
the relationship with decimals
(tenths, hundredths emphasise
connection to tens, units etc) 1then
other
common fractions e.g. 4 =
25
=
025.
100
Need to keep emphasising
equivalent fractions.
Starting with fractions, then teach
the relationship with percentages,
finally link percentages to
decimals.
60
60% = 100
= 06
Pupils need to be secure at finding
common percentages of a quantity,
by linking the percentage to
fractions.
e.g. 1%, 10%, 20%, 25%, 50%,
75% and 100%.
Time
I can use and interpret electronic and paper-based timetables and
schedules to plan events and activities, and make time calculations as
part of my planning.
Term/Definition
Methodology
a.m. - ante meridian
p.m. - post meridian
24 hour time
Speed
Example
Calculating duration.
8:35am4:20pm
8:35am9:00am = 25mins
(9:00am12:00noon = 3h)
12:00noon4:00pm = 4h
4:00pm4:20pm = 20mins
7hours 45minutes
Correct Use of Language
Be aware and teach the various
ways we speak of time.
3:30pm
Analogue – half past three in the
afternoon.
Digital - three thirty pm.
Pupils should be familiar with :
noon; midday; midnight;
afternoon; evening; morning; night
and the different conventions for
recording the date.
Say
zero two hundred hours. (Children
should be aware of different
displays, e.g. 02:00, 02 00 and
0200)
eight kilometres per hour. 8km/h
sixteen metres per second. 16m/s
When calculating the duration
pupils should clearly set out steps
.
Measurement
I can use my knowledge of the sizes of familiar objects or places to assist
me when making an estimate of measure.
I can use the common units of measure, convert between related units of
the metric system and carry out calculations when solving problems
I can explain how different methods can be used to find the perimeter
and area of a simple 2D shape or volume of a simple 3D object.
Example
A1 = lb
= 12 × 4
= 48cm2
A2 = lb
=5×4
= 20cm2
Total Area = A1 + A2
= 48 + 20
= 68cm2
1cm3=1ml
1000cm3 = 1000ml
= 1litre
Correct Use of Language
3cm2
Say 3 square centimetres not 3
centimetres squared or 3 cm two.
Abbreviation of l for litre.
Say 3 litres. (3l)
Abbreviation of ml for millilitres.
Say seven hundred millilitres.
(700ml)
2.30m
543m
6124kg
Pupils should understand how to
write measurements (in m, cm, kg,
g), how to say them and what they
mean e.g. 5 metres 43cm.
Methodology
To find the area of compound
shapes:

Split the shape into
rectangles

Label them as shown

Fill in any missing lengths
12m
A1
9m
A2
4m
4m
Six kilograms and 124 grams, say
six point one two four kilograms.
Emphasise that perimeter is the
distance around the outside of the
shape.
A=lb
Start with this and move to A = lb
when appropriate.
6cm
10cm
DO NOT USE A = 12 l  b or A = 12 lb
as this leads to confusion later on
with the base and height of a triangle.
80 cm3
Complete the surrounding
rectangle if necessary.
Area of rectangle = 10  62
= 60cm
Area of Triangle = 12 the Area of
rectangle
= 12 of 60
= 30cm2
Say 80 cubic centimetres NOT 80
centimetres cubed.
Use litres or millilitres for volume
with liquids.
Use cm3 or m3 for capacity.
Patterns and relationships
Having explored more complex number sequences, including wellknown named number patterns, I can explain the rule used to generate
the sequence, and apply it to extend the pattern.
Term/Definition
Methodology
Prime numbers: numbers with
only 2 factors, one and themselves.
One is not defined as a prime
number.
2, 3, 5, 7, 11, 13, 17, 19, …
See Algebra Appendix
Square numbers
1, 4, 9, 16, 25, … Should be
learned.
Triangular numbers
1, 3, 6, 10, 15… Should be learned.
Example
6, 12, 20, 30, …
Correct Use of Language
Pupils also have to be able to
continue the sequence when the
steps are not constant, but not give
a rule.
Expressions and equations
I can apply my knowledge of number facts to solve problems where an
unknown value is represented by a symbol or letter.
Example
Pupils should be introduced to
single function machines and then
double function machines.
3→21
8→56
10→70
Correct Use of Language
Use “in” and “out” to raise
awareness of “input” and “output.”
Pupils should use the following
terminology:
Input; output; reverse; do the
opposite; work backwards; inverse;
undo etc.
Outputs larger than the input, so
the options are addition or
multiplication
Similarly if the outputs are smaller
it implies subtraction or division.
Methodology
Establish the operations that are an
option.
See Algebra Appendix
Properties of 2D and 3D shapes
Having explored a range of 3D objects and 2D shapes, I can use
mathematical language to describe their properties, and through
investigation can discuss where and why particular shapes are used in the
environment.
Through practical activities, I can show my understanding of the
relationship between 3D objects and their nets.
I can draw 2D shapes and make representations of 3D objects using an
appropriate range of methods and efficient use of resources.
Term/Definition
Congruent: Two shapes are congruent if all the sides are the same length
and all the angles are the same i.e. the shapes are identical.
Example
The faces on a cube are congruent.
Correct Use of Language
Pupils should be familiar with the word congruent.
Angle, symmetry and transformation
Through practical activities which include the use of technology, I have
developed my understanding of the link between compass points and
angles and can describe, follow and record directions, routes and
journeys using appropriate vocabulary.
Term/Definition
060o
Methodology
Say zero six zero degrees.
I can display data in a clear way using a suitable scale, by choosing
appropriately from an extended range of tables, charts, diagrams and
graphs, making effective use of technology.
Term/Definition
Histogram: no spaces between the
bars, unlike a bar graph. (Used to
display grouped data.)
Continuous Data: can have an
infinite number of possible values
within a selected range.
(Temperature, height or length)
Discrete Data: can only have a
finite or limited number of possible
values. (Shoe size, number of
siblings)
Non-numerical data: data which
is non-numerical (Favourite
flavour of crisps)
Use a bar graph, pictogram or pie
chart to display discrete data or
non-numerical data.
Methodology
See Information Handling
Appendix
Appendix 1:
Common Methodology
for Algebra
Common Methodology - Algebra
Overview
Algebra is a way of thinking, i.e. a method of seeing and expressing
relationships, and generalising patterns - it involves active exploration and
conjecture. Algebraic thinking is not the formal manipulation of symbols.
Algebra is not simply a topic that pupils cover in Secondary school. From Primary
One, pupils lay the foundations for algebra. This includes:
Early, First and Second Level



Writing equations e.g. 16 add 8 equals?
Solving equations e.g. 2   = 7
Finding equivalent forms
e.g. 24 = 20 + 4 = 30 – 6
24 = 6  4 = 3  2  2  2


Using inverses or reversing e.g. 4 + 7 = 11→ 11 – 7 = 4
Identifying number patterns
Early/First Level - Language
4 + 5 = 9 is the start of thinking about equations, as it is a statement of equality
between two expressions.
Move from “makes” towards “equals” when concrete material is no longer
necessary. Pupils should become familiar with the different vocabulary for addition
and subtraction as it is encountered.
Second, Third and Fourth Level
Expressing relationships

Drawing graphs

Factorising numbers and expressions

Understanding the commutative, associative and distributive laws
First/Second Level – Recognise and explain simple relationships
Establish the operation(s) that are an option.
The outputs are larger than
the input, so the options are
either addition or
multiplication.
3→21
8→56
10→70
18→9
14→7
6→3
The outputs are smaller than
the input, so the options are
either subtraction or
division.
In this case, outputs are larger
so the options are addition or
multiplication. Add 2 works
for the first one but
2 add 2 gives 4, we need the
answer 6, addition does not
work.
For multiplication, look at
which table the output values
are in. Ensure you check the
answer.
Second Level – Use and devise simple rules
Pupils need to be able to use notation to describe general relationships between 2 sets of numbers, and then use and
devise simple rules.
Pupils need to be able to deal with numbers set out in a table horizontally, set out in a table vertically or given as a
sequence.
A method should be followed, rather than using “trial and error” to establish the rule.
At Second Level pupils have been asked to find the rule by establishing the single operation used. (+5, ×3, ÷2)
Example 1: Complete the following table, finding the nth term.
In this example, the output values are still
increasing, however, addition or multiplication
on their own do not work, so this must be a
two-step operation.
Input
Output
1
5
2
7
3
9
4
11
5
13
n
?
Look at the outputs. These are going up by 2 each time. This tells us that we are multiplying by 2. (This means × 2.)
Now ask:
1 multiplied by 2 is 2, how do we get to 5? Add 3.
2 multiplied by 2 is 4, how do we get to 7? Add 3.
This works, so the rule is:
Multiply by 2 then add 3.
Check using the input 5:
5 × 2 + 3 = 13
We use n to stand for any number
So the nth term would be
n × 2 + 3 which is rewritten as
2n + 3
Example 2: Find the 20th term.
Input
1
2
3
4
5
6
Output
7
10
13
16
19
22
n
3n + 4
+3
+3
To find the 20th term
it is best to find the nth
first.
20
Look at the output values. These are going up by 3 each time. This tells us that we are multiplying by 3. (This means
× 3.)
Now ask:
1 multiplied by 3 is 3, how do we get to 7? Add 4.
2 multiplied by 3 is 6, how do we get to 10? Add 4.
This works so the rule is
Multiply by 3 then add 4.
Check using 6:
We use n to stand for any number
So the nth term would be
6 × 3 + 4 = 22
n × 3 + 4 which is rewritten as
3n + 4
To get the 20th term we substitute n = 20 into our formula.
3n + 4
=3×20 + 4
=60 + 4
=64
So the 20th term is 64.
Example 3:
For the following sequence find the term that produces an output of 90.
Input
1
2
3
4
5
6
Output
2
10
18
26
34
42
n
8n – 6
+8
+8
90
We go through the same process as before to find the nth term, which is 8n – 6.
Now we set up an equation.
8n  6 = 90
+6 +6
8n =96
8 8
n = 12
Therefore the 12th term produces an output of 90.
Second/Third Level – Using formulae
Pupils meet formulae in ‘Area’, ‘Volume’, ‘Circle’, ‘Speed, Distance, Time’ etc. In all
circumstances, working must be shown which clearly demonstrates strategy, (i.e. selecting the
correct formula), substitution and evaluation.
Example :
Find the area of a triangle with base 8cm and height 5cm.
A  12 bh
A  85
1
2
A  20
Area of triangle is 20 cm2
Remember the
importance of the
substitution.
Appendix 2:
Common Methodology
for Information Handling
Information Handling
Discrete Data
Discrete data can only have a finite or limited number of possible values.
Shoe sizes are an example of discrete data because sizes 39 and 40 mean something, but size 39∙2, for example, does
not.
Continuous Data
Continuous data can have an infinite number of possible values within a selected range.
e.g. temperature, height, length.
Non-Numerical Data (Nominal Data)
Data which is non-numerical.
e.g. favourite TV programme, favourite flavour of crisps.
Tally Chart/Table (Frequency table)
A tally chart is used to collect and organise data prior to representing it in a graph.
Averages
Pupils should be aware that mean, median and mode are different types of average.
Mean: add up all the values and divide by the number of values.
Mode: is the value that occurs most often.
Median: is the middle value or the mean of the middle pair of an ordered set of values.
Pupils are introduced to the mean using the word average. In society average is commonly used to refer to the mean.
Range
The difference between the highest and lowest value.
Pictogram/pictograph
A pictogram/pictograph should have a title and appropriate x (horizontal) and y-axis (vertical) labels. If each picture
represents a value of more than one, then a key should be used.
The weight each pupil managed to
lift
ME AN
represents two units
Bar Chart/Graph
A bar chart is a way of displaying discrete or non-numerical data. A bar chart should have a title and appropriate
x and y-axis labels. An even space should be between each bar and each bar should be of an equal width. Leave a
space between the y-axis and the first bar. When using a graduated axis, the intervals must be evenly spaced.
Favourite Fruit
Number of pupils
10
8
6
4
2
0
Plum
Grapes
Banana
Fruit
Apple
1. Histogram
A histogram is a way of displaying grouped data. A histogram should have a title and appropriate x and yaxis labels. There should be no space between each bar. Each bar should be of an equal width. When using a
graduated axis, the intervals must be evenly spaced.
Height of P7 pupils
Number of pupils
8
7
6
5
4
3
2
1
0
130-134 135-139 140-144 145-149 150-154 155-159 160-164 165-169
Height (cm)
Pie Charts
A pie chart is a way of displaying discrete or non-numerical data.
It uses percentages or fractions to compare the data. The whole circle (100% or one whole) is then split up
into sections representing those percentages or fractions. A pie chart needs a title and a key.
S1 pupils favourite sport
10%
Football
Badminton
25%
50%
Swimming
Athletics
Basketball
10%
5%
Line Graphs
Line graphs compare two quantities (or variables). Each variable is plotted along an axis. A line graph has a
vertical and horizontal axis. So, for example, if you wanted to graph the height of a ball after you have
thrown it, you could put time along the horizontal, or x-axis, and height along the vertical, or y-axis.
A line graph needs a title and appropriate x and y-axis labels. If there is more than one line graph on the same
axes, the graph needs a key.
Mental maths result out of 10
A comparison of pupils mental maths results
10
9
8
7
6
5
4
3
2
1
0
John
Katy
1
2
3
4
Week
5
6
Scattergraphs (Scatter diagrams)
A scattergraph allows you to compare two quantities (or variables). Each variable is plotted along an axis. A
scattergraph has a vertical and hrizontal axis. It needs a title and appropriate x and y-axis labels. For each
piece of data a point is plotted on the diagram. The points are not joined up.
A scattergraph allows you to see if there is a connection (correlation) between the two quantities. There may
be a positive correlation when the two quantities increase together e.g. sale of umbrellas and rainfall. There
may be a negative correlation were as one quantity increases the other decreases e.g. price of a car and the
age of the car. There may be no correlation e.g. distance pupils travel to school and pupils’ heights.
Science mark (out of 25)
A comparison of pupils' Maths and Science
marks
25
20
15
10
5
0
0
5
10
15
Maths mark (out of 25)
20
25